Simulation of the droplet-to-bubble transition in a two-fluid system

Recent experiments by Burton and Taborek have demonstrated a droplet-to-bubble transition in the pinchoff behavior of one inviscid fluid inside another. With $D$ the relative densities ${\rho }_{\mathcal{E}}/{\rho }_{\Im }$, they find transition from ($D=0$) droplet-to-bubble behavior at $D\approx 4$. Numerical simulations of this two-fluid system, up to and beyond the initial breakup of the inner fluid, have been carried out utilizing level set and boundary integral methods. A droplet-to-bubble transition is predicted: For $D$ sufficiently large, the volume of the satellite droplet shrinks to zero and there is no overturning of the fluid at separation. The calculated self-similar scaling exponents and the pinchoff region shapes match the known behavior at the droplet and bubble extremes ($D=0$, $D=100$). For intermediate $D$ values, the simulations presented here indicate that the transition range between droplet and bubble behavior depends upon initial drop geometry. When the neck separates two nonequal inner fluid masses the transition is mild and occurs in the range $4<D<10$, whereas in the case of equal masses an abrupt transition occurs at $D\approx 4$ in perfect agreement with the above mentioned experiments.

Figure 1

Physical setting in three dimensions.

Figure 2

Computational domain in the $r$-$z$ plane, drop geometry 1.

Figure 3

Computational domain in the $r$-$z$ plane, drop geometry 2.

Figure 4

Minimum neck radius time evolution for $D=0,1,2,4,6,20,100$ (left to right).

Figure 5

Focused fronts at pinchoff.

Figure 6

Front profiles for $D=6$, coarser and finer grids.

Figure 7

Focused front profiles for $D=4$, coarser and finer grids.

Figure 8

Focused front profiles for $D=6$, coarser and finer grids.

Figure 9

Front focused at $t=0.635$ for $D=6$ showing BEM nodes.

Figure 10

Front focused at $t=1.897\hspace{0.5em}14$ for $D=100$ showing BEM nodes.

Figure 11

Front profiles at indicated times, $D=0$.

Figure 12

Front profiles at indicated times, $D=0.3$.

Figure 13

Front profiles at indicated times, $D=1$.

Figure 14

3D Front profiles at various times (same as in Fig. 13), $D=1$. For the horizontal axis $0.4⩽z⩽6.6$ and the vertical axis $-1.33⩽r⩽1.33$.

Figure 15

Front profiles at indicated times, $D=2$.

Figure 16

Front profiles at indicated times, $D=4$.

Figure 17

Front profiles at indicated times, $D=6$.

Figure 18

Front profiles at indicated times, $D=10$.

Figure 19

Front profiles at indicated times, $D=100$.

Figure 20

Focused details of satellite pinching and coalescing at indicated times, $D=0.3$.

Figure 21

Scaling of $r_{\min }$ at pinchoff for $D=0$ and $D=100$.

Figure 22

Scaling of $r_{\min }$ for $D=4$ (squares for the fitted region).

Figure 23

Scaling of $r_{\min }$ for $D=6$ (squares for the fitted region).

Figure 24

Scaling exponents versus $D$, calculated and experimental values, geometry 1.

Figure 25

Scaling exponents versus $D$, calculated and experimental values, geometry 2.

Figure 26

$r_{\min }$ evolution, theoretical and calculated, for $D=6$ and geometry 2.

Figure 27

3D profiles at pinchoff for $D=0,1,2,4,6$ (top to bottom), geometry $2$. For the horizontal axis $0.4⩽z⩽6.6$ and the vertical axis $-1.33⩽r⩽1.33$.

Figure 28

3D focused profiles at pinchoff for $D=0,1,2$ (top to bottom), ${30}^{°}$ view.

Figure 29

3D focused profiles at pinchoff for $D=4,6$ (top to bottom), front view. For the horizontal axis $2.9⩽z⩽4.1$ and the vertical axis $-0.26⩽r⩽0.26$.

Figure 30

3D profiles after pinchoff for $D=4$, global and focused. For the horizontal axis $2.9⩽z⩽4.1$ and the vertical axis $-0.26⩽r⩽0.26$.

Figure 31

Satellite volume versus $D$, theoretical and calculated, geometry 1.